Marginal Effects

02-05

Marginal Effects
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As Cameron & Trivedinote (p. 333), 「An ME [marginal effect], or partial effect, most often measures the effect on the conditional mean of y of a change in one of the regressors, say Xk. In the linear regression model, the ME equals the relevant slope coefficient, greatly simplifying analysis. For nonlinear models, this is no longer the case, leading to remarkably many different methods for calculating MEs.」

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Marginal effects are popular in some disciplines (e.g. Economics) because they often provide a good approximation to the amount of change in Y that will be produced by a 1-unit change in Xk. With binary dependent variables, they offer some of the same advantages that the Linear Probability Model (LPM) does –they give you a single number that expresses the effect of a variable on P(Y=1).

進行回歸分析往往要看邊際影響，對於線性模型邊際影響就是其係數

但對於許多非線性模型邊際影響是不等於係數值的，例如：logit、probit

並且「More common are approaches which focus on discrete changes」

Discrete Change for Categorical Variables.

The MEM for categorical variables therefore shows how P(Y=1) changes as the categorical variable changes from 0 to 1, holding all other variables at their means. That is, for a categorical variable Xk:

Marginal Effect Xk = Pr(Y = 1|X, Xk = 1) – Pr(y=1|X, Xk = 0)

結果告訴我們，如果你有兩個其他的平均個體，一個白人，一個黑人，那麼黑人糖尿病的概率將會高出2.9個百分點 (Black APM = .0585, white APM = .0294, MEM = .0585 -.0294 = .029).
And what do we mean by average?
使用APMs和MEMs，平均值被定義為模型中其他自變數的平均值，即47.57歲，黑人10.5％，女性52.5％。

然而對於連續變數的應用並不多：

When Xk is measured in small units, e.g. income in dollars, the effect of a 1 unit increase in Xk may match up well with the MEM for Xk. But, when Xk is measured in larger units (e.g. income in millions of dollars) the MEM may or may not provide a very good approximation of the effect of a one unit increase in Xk. That is probably one reason why instantaneous rates of change for continuous variables receive relatively little attention, at least in Sociology.